Failure of the Singularity Theorems on Discrete Spacetimes: Curvature Bounds and Geodesic Completeness with Minimum Length

For sixty years, the Hawking-Penrose singularity theorems have stood as one of the most profound results in theoretical physics: under generic conditions, any spacetime satisfying Einstein's equations must contain singularities—points where curvature becomes infinite and the laws of physics break down. These theorems established that singularities are not mathematical artifacts but inevitable consequences of general relativity itself. Yet singularities are also the theory's greatest failure, signaling that the classical description must break down at the Planck scale.

This paper proves that the singularity theorems do not extend to spacetimes with a fundamental minimum length. On a simplicial spacetime with minimum edge length, three results are established. First, a universal curvature bound: all curvature invariants are bounded by a constant divided by the square of the minimum length. Second, geodesic completeness: every causal geodesic can be extended to arbitrary affine parameter—no paths terminate in singularities. Third, a detailed examination of the Hawking-Penrose conditions shows that while trapped surfaces, energy conditions, and global hyperbolicity can all be satisfied on a discrete spacetime, they do not force geodesic incompleteness. The crucial step in the continuum proof—that a trapped surface implies infinite focusing of geodesics in finite affine parameter—fails when curvature is bounded. The focusing saturates at the curvature bound rather than diverging.

The result establishes that classical singularities are artifacts of the continuum idealization. In any theory with a minimum length, spacetime is necessarily geodesically complete. Black hole interiors are regions of Planckian curvature, not singularities. The Big Bang is replaced by a finite-size initial state. No singularities form, regardless of matter content or initial conditions.

This work is extracted from the Emergence model, a unified framework in which spacetime is a discrete lattice formed from wave intersections on a pre-geometric canvas. The proof relies on the Regge calculus convergence results and the well-posedness of discrete particle-field dynamics established in companion papers.